Dynamical model for the polar-incommensurate transition in BaMnF4

Abstract

Using a Landau-Khalatnikov approach we calculate the temperature dependence of the dielectric constant and of the coupled normal modes in incommensurate phase transitions such as that displayed by BaMn${\mathrm{F}}_{4}$. It is assumed that the phase mode is relaxational in the incommensurate phase and satisfies a Debye relaxation equation. Two results of interest are generated: first, the temperature dependence of the inverse relaxation time is given by ${\ensuremath{\tau}}^{\ensuremath{-}1}(T)=[\frac{({T}_{\mathrm{I}}\ensuremath{-}T)}{{T}_{\mathrm{I}}}]{\ensuremath{\tau}}_{0}^{\ensuremath{-}1}+{\ensuremath{\tau}}_{1}^{\ensuremath{-}1}$, in agreement with experimental results for both BaMn${\mathrm{F}}_{4}$ and ${\mathrm{Ba}}_{2}$Na${\mathrm{Nb}}_{5}$${\mathrm{O}}_{15}$; and second, a new relationship between the dynamical parameters ${\ensuremath{\tau}}_{0}$ and ${\ensuremath{\tau}}_{1}$ and the temperature width of stability for the incommensurate phase, ${T}_{\mathrm{I}}\ensuremath{-}{T}_{\mathrm{II}}$ is derived: $\frac{({T}_{\mathrm{I}}\ensuremath{-}{T}_{\mathrm{II}})}{{T}_{\mathrm{I}}}=C\frac{{\ensuremath{\tau}}_{0}}{{\ensuremath{\tau}}_{1}}$, where $C$ is a numerical constant of order 1.0 to 6.0 in the anisotropic case. Using experimental values of $\frac{1}{2}\ensuremath{\pi}{\ensuremath{\tau}}_{0}=(1.7\ifmmode\pm\else\textpm\fi{}0.7)\ifmmode\times\else\texttimes\fi{}{10}^{11}$ Hz and $\frac{1}{2}\ensuremath{\pi}{\ensuremath{\tau}}_{1}=(6.7\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}{10}^{8}$ Hz for BaMn${\mathrm{F}}_{4}$, we calculate ${T}_{\mathrm{I}}\ensuremath{-}{T}_{\mathrm{II}}=6\ifmmode\pm\else\textpm\fi{}4$ K, in good agreement with the recent measurement of 8.2 K by Scott, Habbal, and Hidaka; similar estimates for ${\mathrm{Ba}}_{2}$Na${\mathrm{Nb}}_{5}$${\mathrm{O}}_{15}$ predict a large incommensurate temperature region, which is also in good accord with experiment.